Looking at the data

We plot the data and can see that there is no obvious large difference between the debt versions

d.both_completed %>%
  ggplot(aes(x=time/60, fill=high_debt_version)) + 
  geom_boxplot() +
  labs(
    title = "Distribution of time measurements for the different debt levels",
    subtitle = "Notice! Log10 x-scale",
    x ="Time (min)"
  ) +
  scale_y_continuous(breaks = NULL) +
  scale_x_log10() +
  scale_fill_discrete(name = "Debt Level", labels = c("High Debt", "Low Debt"), guide = guide_legend(reverse = TRUE)) +
  theme_minimal()

Descriptive Statistics

For the whole dataset:

d.both_completed %>%
  pull(time) %>% 
  summary()
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   442.0   937.5  1358.0  1842.2  2423.8  7298.0
variance <- d.both_completed %>%
  pull(time) %>%
  var()
sprintf("Variance: %.0f", variance)
## [1] "Variance: 1880646"

Initial model

As the variance is much greater than the mean we will use a negative binomial family that allows us to model the variance separately.

We include high_debt_verison as well as a varying intercept for each individual in our initial model.

Selecting Priors

We iterate over the model until we have sane priors

time0.with <- extendable_model(
  base_name = "time0",
  base_formula = "time ~ 1 + high_debt_version + (1 | session)",
  base_priors = c(
    prior(normal(0, 1), class = "b"),
    prior(normal(7.5, 1), class = "Intercept"),
    prior(exponential(1), class = "sd"),
    prior(gamma(0.01, 0.01), class = "shape")
  ),
  family = negbinomial(),
  data = d.both_completed,
  base_control = list(adapt_delta = 0.95)
)

# Default priors:
prior_summary(time0.with(only_priors= TRUE))
# Our priors:
prior_summary(time0.with(sample_prior = "only"))
# Prior predictive check
pp_check(time0.with(sample_prior = "only"), nsamples = 200) + 
  scale_x_log10()
## Warning: Transformation introduced infinite values in continuous x-axis
## Warning: Removed 8171 rows containing non-finite values (stat_density).
## Warning: Groups with fewer than two data points have been dropped.

## Warning: Groups with fewer than two data points have been dropped.
## Warning: Removed 2 row(s) containing missing values (geom_path).

### Model fit We check the posterior distribution and can see that the model seems to have been able to fit the data well

# Posterior predictive check
pp_check(time0.with(), nsamples = 100) +
  scale_x_log10()

summary(time0.with())
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: time ~ 1 + high_debt_version + (1 | session) 
##    Data: as.data.frame(data) (Number of observations: 44) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~session (Number of levels: 22) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.42      0.15     0.08     0.73 1.00      833      801
## 
## Population-Level Effects: 
##                        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept                  7.53      0.15     7.24     7.85 1.00     2704
## high_debt_versionfalse    -0.18      0.17    -0.51     0.16 1.00     4532
##                        Tail_ESS
## Intercept                  2564
## high_debt_versionfalse     3054
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     3.64      1.06     1.97     6.04 1.00     1215     1892
## 
## Samples were drawn using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Model extenstions

We use loo to check some possible extensions on the model.

edlvl_prior <- prior(dirichlet(2), class = "simo", coef = "moeducation_level1")

loo(
  time0.with(),
  time0.with("work_domain"),
  time0.with("work_experience_programming.s"),
  time0.with("work_experience_java.s"),
  time0.with("education_field"),
  time0.with("mo(education_level)", edlvl_prior),
  time0.with("workplace_peer_review"),
  time0.with("workplace_td_tracking"),
  time0.with("workplace_pair_programming"),
  time0.with("workplace_coding_standards"),
  time0.with("scenario"),
  time0.with("group")
)
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.9 seconds.
## Total execution time: 1.2 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.7 seconds.
## Total execution time: 0.9 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.7 seconds.
## Total execution time: 1.0 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.8 seconds.
## Total execution time: 1.0 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## Start sampling
## Running MCMC with 4 parallel chains...
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## Warning: Found 4 observations with a pareto_k > 0.7 in model 'time0.with()'. It
## is recommended to set 'moment_match = TRUE' in order to perform moment matching
## for problematic observations.
## Warning: Found 6 observations with a pareto_k > 0.7 in model
## 'time0.with("work_domain")'. It is recommended to set 'moment_match = TRUE' in
## order to perform moment matching for problematic observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with("work_experience_programming.s")'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("work_experience_java.s")'. It is recommended to set 'moment_match =
## TRUE' in order to perform moment matching for problematic observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("education_field")'. It is recommended to set 'moment_match = TRUE'
## in order to perform moment matching for problematic observations.
## Warning: Found 4 observations with a pareto_k > 0.7 in model
## 'time0.with("mo(education_level)", edlvl_prior)'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("workplace_peer_review")'. It is recommended to set 'moment_match =
## TRUE' in order to perform moment matching for problematic observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("workplace_td_tracking")'. It is recommended to set 'moment_match =
## TRUE' in order to perform moment matching for problematic observations.
## Warning: Found 4 observations with a pareto_k > 0.7 in model
## 'time0.with("workplace_pair_programming")'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("workplace_coding_standards")'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with("scenario")'. It is recommended to set 'moment_match = TRUE' in
## order to perform moment matching for problematic observations.
## Warning: Found 8 observations with a pareto_k > 0.7 in model
## 'time0.with("group")'. It is recommended to set 'moment_match = TRUE' in order
## to perform moment matching for problematic observations.
## Output of model 'time0.with()':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.5  7.3
## p_loo        14.5  3.0
## looic       734.9 14.6
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   735       
##  (0.5, 0.7]   (ok)       12    27.3%   556       
##    (0.7, 1]   (bad)       4     9.1%   38        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("work_domain")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -370.6  6.8
## p_loo        17.7  2.9
## looic       741.2 13.7
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     25    56.8%   753       
##  (0.5, 0.7]   (ok)       13    29.5%   384       
##    (0.7, 1]   (bad)       4     9.1%   33        
##    (1, Inf)   (very bad)  2     4.5%   23        
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("work_experience_programming.s")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.3  7.1
## p_loo        15.1  2.9
## looic       736.5 14.2
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     30    68.2%   769       
##  (0.5, 0.7]   (ok)        9    20.5%   544       
##    (0.7, 1]   (bad)       5    11.4%   52        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("work_experience_java.s")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.0  7.0
## p_loo        14.9  2.7
## looic       735.9 13.9
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     27    61.4%   643       
##  (0.5, 0.7]   (ok)       14    31.8%   333       
##    (0.7, 1]   (bad)       3     6.8%   76        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("education_field")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.2  6.2
## p_loo        13.1  2.1
## looic       734.5 12.4
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     25    56.8%   762       
##  (0.5, 0.7]   (ok)       16    36.4%   337       
##    (0.7, 1]   (bad)       3     6.8%   74        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("mo(education_level)", edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.0  6.7
## p_loo        12.6  2.4
## looic       734.0 13.3
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     26    59.1%   734       
##  (0.5, 0.7]   (ok)       14    31.8%   612       
##    (0.7, 1]   (bad)       4     9.1%   46        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("workplace_peer_review")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.2  6.7
## p_loo        13.4  2.6
## looic       736.5 13.4
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     29    65.9%   665       
##  (0.5, 0.7]   (ok)       12    27.3%   301       
##    (0.7, 1]   (bad)       2     4.5%   21        
##    (1, Inf)   (very bad)  1     2.3%   56        
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("workplace_td_tracking")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.9  8.1
## p_loo        16.2  3.9
## looic       737.7 16.2
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     33    75.0%   383       
##  (0.5, 0.7]   (ok)        8    18.2%   699       
##    (0.7, 1]   (bad)       1     2.3%   314       
##    (1, Inf)   (very bad)  2     4.5%   4         
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("workplace_pair_programming")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.6  6.9
## p_loo        14.5  2.6
## looic       735.1 13.7
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     30    68.2%   851       
##  (0.5, 0.7]   (ok)       10    22.7%   386       
##    (0.7, 1]   (bad)       4     9.1%   110       
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("workplace_coding_standards")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.4  6.9
## p_loo        14.8  2.7
## looic       736.7 13.8
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   506       
##  (0.5, 0.7]   (ok)       13    29.5%   127       
##    (0.7, 1]   (bad)       3     6.8%   37        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("scenario")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -366.0  7.8
## p_loo        16.9  3.4
## looic       732.0 15.5
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     25    56.8%   749       
##  (0.5, 0.7]   (ok)       14    31.8%   370       
##    (0.7, 1]   (bad)       5    11.4%   40        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("group")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -369.7  7.0
## p_loo        16.3  3.0
## looic       739.4 14.0
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     24    54.5%   674       
##  (0.5, 0.7]   (ok)       12    27.3%   704       
##    (0.7, 1]   (bad)       8    18.2%   19        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Model comparisons:
##                                                elpd_diff se_diff
## time0.with("scenario")                          0.0       0.0   
## time0.with("mo(education_level)", edlvl_prior) -1.0       3.4   
## time0.with("education_field")                  -1.2       3.5   
## time0.with()                                   -1.5       2.1   
## time0.with("workplace_pair_programming")       -1.6       2.3   
## time0.with("work_experience_java.s")           -2.0       2.1   
## time0.with("workplace_peer_review")            -2.2       2.5   
## time0.with("work_experience_programming.s")    -2.3       2.2   
## time0.with("workplace_coding_standards")       -2.4       2.3   
## time0.with("workplace_td_tracking")            -2.9       2.0   
## time0.with("group")                            -3.7       2.1   
## time0.with("work_domain")                      -4.6       3.4

Some variables seem to stand out as better than other. We try combining those to see if we can further improve our model.

loo(
  time0.with(),
  time0.with("scenario"),
  time0.with("education_field"),
  time0.with("workplace_peer_review"),
  time0.with("mo(education_level)", edlvl_prior),
  time0.with(c("scenario", "education_field")),
  time0.with(c("scenario", "mo(education_level)"), edlvl_prior),
  time0.with(c("scenario", "workplace_peer_review")),
  time0.with(c("education_field", "mo(education_level)"), edlvl_prior),
  time0.with(c("education_field", "workplace_peer_review")),
  time0.with(c("mo(education_level)", "workplace_peer_review"), edlvl_prior)
)
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.9 seconds.
## Total execution time: 1.2 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 2.2 seconds.
## Total execution time: 2.7 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## Chain 1 finished in 0.9 seconds.
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## Chain 4 finished in 0.9 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.9 seconds.
## Total execution time: 1.0 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## Chain 4 finished in 2.5 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 2.3 seconds.
## Total execution time: 2.6 seconds.
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
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## Mean chain execution time: 0.9 seconds.
## Total execution time: 1.1 seconds.
## 
## Warning: 2 of 4000 (0.0%) transitions ended with a divergence.
## This may indicate insufficient exploration of the posterior distribution.
## Possible remedies include: 
##   * Increasing adapt_delta closer to 1 (default is 0.8) 
##   * Reparameterizing the model (e.g. using a non-centered parameterization)
##   * Using informative or weakly informative prior distributions 
## 
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
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## Mean chain execution time: 2.1 seconds.
## Total execution time: 2.3 seconds.
## Warning: Found 4 observations with a pareto_k > 0.7 in model 'time0.with()'. It
## is recommended to set 'moment_match = TRUE' in order to perform moment matching
## for problematic observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with("scenario")'. It is recommended to set 'moment_match = TRUE' in
## order to perform moment matching for problematic observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("education_field")'. It is recommended to set 'moment_match = TRUE'
## in order to perform moment matching for problematic observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model
## 'time0.with("workplace_peer_review")'. It is recommended to set 'moment_match =
## TRUE' in order to perform moment matching for problematic observations.
## Warning: Found 4 observations with a pareto_k > 0.7 in model
## 'time0.with("mo(education_level)", edlvl_prior)'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with(c("scenario", "education_field"))'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with(c("scenario", "mo(education_level)"), edlvl_prior)'. It is
## recommended to set 'moment_match = TRUE' in order to perform moment matching for
## problematic observations.
## Warning: Found 4 observations with a pareto_k > 0.7 in model
## 'time0.with(c("scenario", "workplace_peer_review"))'. It is recommended to
## set 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with(c("education_field", "mo(education_level)"), edlvl_prior)'. It is
## recommended to set 'moment_match = TRUE' in order to perform moment matching for
## problematic observations.
## Warning: Found 4 observations with a pareto_k > 0.7 in model
## 'time0.with(c("education_field", "workplace_peer_review"))'. It is recommended
## to set 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 4 observations with a pareto_k > 0.7 in model
## 'time0.with(c("mo(education_level)", "workplace_peer_review"), edlvl_prior)'. It
## is recommended to set 'moment_match = TRUE' in order to perform moment matching
## for problematic observations.
## Output of model 'time0.with()':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.5  7.3
## p_loo        14.5  3.0
## looic       734.9 14.6
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   735       
##  (0.5, 0.7]   (ok)       12    27.3%   556       
##    (0.7, 1]   (bad)       4     9.1%   38        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("scenario")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -366.0  7.8
## p_loo        16.9  3.4
## looic       732.0 15.5
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     25    56.8%   749       
##  (0.5, 0.7]   (ok)       14    31.8%   370       
##    (0.7, 1]   (bad)       5    11.4%   40        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("education_field")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.2  6.2
## p_loo        13.1  2.1
## looic       734.5 12.4
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     25    56.8%   762       
##  (0.5, 0.7]   (ok)       16    36.4%   337       
##    (0.7, 1]   (bad)       3     6.8%   74        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("workplace_peer_review")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.2  6.7
## p_loo        13.4  2.6
## looic       736.5 13.4
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     29    65.9%   665       
##  (0.5, 0.7]   (ok)       12    27.3%   301       
##    (0.7, 1]   (bad)       2     4.5%   21        
##    (1, Inf)   (very bad)  1     2.3%   56        
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("mo(education_level)", edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.0  6.7
## p_loo        12.6  2.4
## looic       734.0 13.3
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     26    59.1%   734       
##  (0.5, 0.7]   (ok)       14    31.8%   612       
##    (0.7, 1]   (bad)       4     9.1%   46        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("scenario", "education_field"))':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -365.3  6.8
## p_loo        15.4  2.5
## looic       730.6 13.5
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   549       
##  (0.5, 0.7]   (ok)       11    25.0%   180       
##    (0.7, 1]   (bad)       5    11.4%   62        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("scenario", "mo(education_level)"), edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -365.9  7.4
## p_loo        15.8  3.1
## looic       731.9 14.8
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     27    61.4%   549       
##  (0.5, 0.7]   (ok)       12    27.3%   370       
##    (0.7, 1]   (bad)       5    11.4%   21        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("scenario", "workplace_peer_review"))':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -366.6  7.2
## p_loo        16.4  3.1
## looic       733.1 14.4
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     26    59.1%   457       
##  (0.5, 0.7]   (ok)       14    31.8%   204       
##    (0.7, 1]   (bad)       4     9.1%   16        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("education_field", "mo(education_level)"), edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.9  6.2
## p_loo        13.5  2.1
## looic       735.8 12.5
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     27    61.4%   534       
##  (0.5, 0.7]   (ok)       12    27.3%   472       
##    (0.7, 1]   (bad)       5    11.4%   69        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("education_field", "workplace_peer_review"))':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.0  5.8
## p_loo        13.0  1.9
## looic       735.9 11.7
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   463       
##  (0.5, 0.7]   (ok)       12    27.3%   265       
##    (0.7, 1]   (bad)       3     6.8%   41        
##    (1, Inf)   (very bad)  1     2.3%   123       
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("mo(education_level)", "workplace_peer_review"),     edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -368.1  6.7
## p_loo        13.2  2.5
## looic       736.1 13.3
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     26    59.1%   517       
##  (0.5, 0.7]   (ok)       14    31.8%   394       
##    (0.7, 1]   (bad)       3     6.8%   100       
##    (1, Inf)   (very bad)  1     2.3%   30        
## See help('pareto-k-diagnostic') for details.
## 
## Model comparisons:
##                                                                                elpd_diff
## time0.with(c("scenario", "education_field"))                                    0.0     
## time0.with(c("scenario", "mo(education_level)"), edlvl_prior)                  -0.6     
## time0.with("scenario")                                                         -0.7     
## time0.with(c("scenario", "workplace_peer_review"))                             -1.3     
## time0.with("mo(education_level)", edlvl_prior)                                 -1.7     
## time0.with("education_field")                                                  -1.9     
## time0.with()                                                                   -2.2     
## time0.with(c("education_field", "mo(education_level)"), edlvl_prior)           -2.6     
## time0.with(c("education_field", "workplace_peer_review"))                      -2.7     
## time0.with(c("mo(education_level)", "workplace_peer_review"),     edlvl_prior) -2.8     
## time0.with("workplace_peer_review")                                            -3.0     
##                                                                                se_diff
## time0.with(c("scenario", "education_field"))                                    0.0   
## time0.with(c("scenario", "mo(education_level)"), edlvl_prior)                   2.2   
## time0.with("scenario")                                                          2.5   
## time0.with(c("scenario", "workplace_peer_review"))                              2.5   
## time0.with("mo(education_level)", edlvl_prior)                                  3.1   
## time0.with("education_field")                                                   2.5   
## time0.with()                                                                    3.1   
## time0.with(c("education_field", "mo(education_level)"), edlvl_prior)            2.7   
## time0.with(c("education_field", "workplace_peer_review"))                       2.6   
## time0.with(c("mo(education_level)", "workplace_peer_review"),     edlvl_prior)  3.2   
## time0.with("workplace_peer_review")                                             3.2

None one of the new models stand out as particularly better than the old models, We try combining the so far most prominent features in one model.

loo(
  time0.with(),
  time0.with("scenario"),
  time0.with(c("scenario", "education_field")),
  time0.with(c("scenario", "mo(education_level)"), edlvl_prior),
  time0.with(c("scenario", "mo(education_level)", "education_field"), edlvl_prior)
)
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
## All 4 chains finished successfully.
## Mean chain execution time: 2.4 seconds.
## Total execution time: 2.7 seconds.
## Warning: Found 4 observations with a pareto_k > 0.7 in model 'time0.with()'. It
## is recommended to set 'moment_match = TRUE' in order to perform moment matching
## for problematic observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with("scenario")'. It is recommended to set 'moment_match = TRUE' in
## order to perform moment matching for problematic observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with(c("scenario", "education_field"))'. It is recommended to set
## 'moment_match = TRUE' in order to perform moment matching for problematic
## observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with(c("scenario", "mo(education_level)"), edlvl_prior)'. It is
## recommended to set 'moment_match = TRUE' in order to perform moment matching for
## problematic observations.
## Warning: Found 5 observations with a pareto_k > 0.7 in model
## 'time0.with(c("scenario", "mo(education_level)", "education_field"),
## edlvl_prior)'. It is recommended to set 'moment_match = TRUE' in order to
## perform moment matching for problematic observations.
## Output of model 'time0.with()':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.5  7.3
## p_loo        14.5  3.0
## looic       734.9 14.6
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   735       
##  (0.5, 0.7]   (ok)       12    27.3%   556       
##    (0.7, 1]   (bad)       4     9.1%   38        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with("scenario")':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -366.0  7.8
## p_loo        16.9  3.4
## looic       732.0 15.5
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     25    56.8%   749       
##  (0.5, 0.7]   (ok)       14    31.8%   370       
##    (0.7, 1]   (bad)       5    11.4%   40        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("scenario", "education_field"))':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -365.3  6.8
## p_loo        15.4  2.5
## looic       730.6 13.5
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   549       
##  (0.5, 0.7]   (ok)       11    25.0%   180       
##    (0.7, 1]   (bad)       5    11.4%   62        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("scenario", "mo(education_level)"), edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -365.9  7.4
## p_loo        15.8  3.1
## looic       731.9 14.8
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     27    61.4%   549       
##  (0.5, 0.7]   (ok)       12    27.3%   370       
##    (0.7, 1]   (bad)       5    11.4%   21        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0.with(c("scenario", "mo(education_level)", "education_field"),     edlvl_prior)':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -366.5  6.9
## p_loo        15.9  2.6
## looic       733.0 13.7
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     23    52.3%   414       
##  (0.5, 0.7]   (ok)       16    36.4%   267       
##    (0.7, 1]   (bad)       4     9.1%   103       
##    (1, Inf)   (very bad)  1     2.3%   23        
## See help('pareto-k-diagnostic') for details.
## 
## Model comparisons:
##                                                                                      elpd_diff
## time0.with(c("scenario", "education_field"))                                          0.0     
## time0.with(c("scenario", "mo(education_level)"), edlvl_prior)                        -0.6     
## time0.with("scenario")                                                               -0.7     
## time0.with(c("scenario", "mo(education_level)", "education_field"),     edlvl_prior) -1.2     
## time0.with()                                                                         -2.2     
##                                                                                      se_diff
## time0.with(c("scenario", "education_field"))                                          0.0   
## time0.with(c("scenario", "mo(education_level)"), edlvl_prior)                         2.2   
## time0.with("scenario")                                                                2.5   
## time0.with(c("scenario", "mo(education_level)", "education_field"),     edlvl_prior)  1.1   
## time0.with()                                                                          3.1

Candidate models

We inspect some of our top performing models.

All models seems to have sampled nicely (rhat = 1 and fluffy plots) they also have about the same fit to the data end similar estimated for the high_debt_version beta parameter

time0 <- time0.with()
summary(time0)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: time ~ 1 + high_debt_version + (1 | session) 
##    Data: as.data.frame(data) (Number of observations: 44) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~session (Number of levels: 22) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.42      0.15     0.08     0.73 1.00      833      801
## 
## Population-Level Effects: 
##                        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept                  7.53      0.15     7.24     7.85 1.00     2704
## high_debt_versionfalse    -0.18      0.17    -0.51     0.16 1.00     4532
##                        Tail_ESS
## Intercept                  2564
## high_debt_versionfalse     3054
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     3.64      1.06     1.97     6.04 1.00     1215     1892
## 
## Samples were drawn using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
ranef(time0)
## $session
## , , Intercept
## 
##                             Estimate Est.Error         Q2.5     Q97.5
## 6033d69a5af2c702367b3a95 -0.10397124 0.2923277 -0.696223825 0.4966043
## 6033d90a5af2c702367b3a96  0.30284461 0.2894601 -0.208266425 0.9025537
## 6034fc165af2c702367b3a98  0.31667613 0.2943956 -0.205337550 0.9436167
## 603500725af2c702367b3a99 -0.49951258 0.3748742 -1.260279000 0.1651044
## 603f97625af2c702367b3a9d -0.17794550 0.2937348 -0.760886600 0.3846346
## 603fd5d95af2c702367b3a9e  0.23315453 0.2918738 -0.281100675 0.8370111
## 60409b7b5af2c702367b3a9f  0.36103269 0.2937580 -0.148411525 0.9932973
## 604b82b5a7718fbed181b336 -0.32355853 0.3259309 -0.979740100 0.2850628
## 6050c1bf856f36729d2e5218  0.11257794 0.2865656 -0.429758800 0.7083303
## 6050e1e7856f36729d2e5219  0.07112404 0.2806434 -0.474379750 0.6792155
## 6055fdc6856f36729d2e521b -0.08112716 0.2890383 -0.644553200 0.5105831
## 60589862856f36729d2e521f  0.04177085 0.2782209 -0.487416150 0.6217947
## 605afa3a856f36729d2e5222 -0.07176116 0.2788774 -0.616067000 0.5016286
## 605c8bc6856f36729d2e5223 -0.35701918 0.3333509 -1.045979750 0.2328373
## 605f3f2d856f36729d2e5224 -0.11611949 0.2764184 -0.676053150 0.4340249
## 605f46c3856f36729d2e5225 -0.24234922 0.3038210 -0.858150400 0.3353478
## 60605337856f36729d2e5226  0.22223325 0.2884269 -0.282401175 0.8308570
## 60609ae6856f36729d2e5228  0.31238482 0.2999458 -0.210652875 0.9414496
## 6061ce91856f36729d2e522e -0.03605460 0.2777166 -0.576248875 0.5414941
## 6061f106856f36729d2e5231 -0.36046023 0.3317783 -1.034268500 0.2408228
## 6068ea9f856f36729d2e523e  0.65907656 0.3483501  0.004251304 1.3611792
## 6075ab05856f36729d2e5247 -0.29447645 0.3227753 -0.937050450 0.2935324
plot(time0, ask = FALSE)

pp_check(time0, nsamples = 150) + scale_x_log10()

time1 <- time0.with("scenario")
summary(time1)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: time ~ 1 + high_debt_version + (1 | session) + scenario 
##    Data: as.data.frame(data) (Number of observations: 44) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~session (Number of levels: 22) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.46      0.14     0.17     0.74 1.00     1012      832
## 
## Population-Level Effects: 
##                        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept                  7.66      0.17     7.33     7.98 1.00     2705
## high_debt_versionfalse    -0.15      0.16    -0.45     0.19 1.00     4024
## scenariotickets           -0.30      0.16    -0.62     0.02 1.00     5218
##                        Tail_ESS
## Intercept                  2629
## high_debt_versionfalse     2523
## scenariotickets            2917
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     4.21      1.27     2.17     7.02 1.00     1091     1575
## 
## Samples were drawn using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
ranef(time1)
## $session
## , , Intercept
## 
##                              Estimate Est.Error       Q2.5     Q97.5
## 6033d69a5af2c702367b3a95 -0.130013228 0.2933120 -0.7014096 0.4636022
## 6033d90a5af2c702367b3a96  0.333815216 0.2884059 -0.1822406 0.9451902
## 6034fc165af2c702367b3a98  0.303688245 0.2842743 -0.2174191 0.8950870
## 603500725af2c702367b3a99 -0.587756292 0.3718096 -1.2880072 0.1032327
## 603f97625af2c702367b3a9d -0.181772348 0.3015216 -0.7529205 0.4268712
## 603fd5d95af2c702367b3a9e  0.211574698 0.2848139 -0.3226416 0.8082725
## 60409b7b5af2c702367b3a9f  0.432166128 0.2931942 -0.1046698 1.0395003
## 604b82b5a7718fbed181b336 -0.379695676 0.3246649 -1.0112815 0.2347585
## 6050c1bf856f36729d2e5218  0.173133529 0.2753742 -0.3473347 0.7506933
## 6050e1e7856f36729d2e5219  0.083091985 0.2830836 -0.4529753 0.6554464
## 6055fdc6856f36729d2e521b -0.070105770 0.2921916 -0.6570595 0.5098490
## 60589862856f36729d2e521f  0.003400744 0.2891862 -0.5625003 0.6001297
## 605afa3a856f36729d2e5222 -0.095780238 0.2857351 -0.6673658 0.4730345
## 605c8bc6856f36729d2e5223 -0.422071106 0.3340411 -1.0848807 0.2101535
## 605f3f2d856f36729d2e5224 -0.084239367 0.2851146 -0.6318389 0.4879983
## 605f46c3856f36729d2e5225 -0.280684904 0.3021689 -0.8841729 0.2999856
## 60605337856f36729d2e5226  0.255534415 0.2840629 -0.2801056 0.8438403
## 60609ae6856f36729d2e5228  0.352847975 0.2850472 -0.1721383 0.9491022
## 6061ce91856f36729d2e522e -0.057323597 0.2849547 -0.6078756 0.5107459
## 6061f106856f36729d2e5231 -0.391315527 0.3370148 -1.0421655 0.2675452
## 6068ea9f856f36729d2e523e  0.797582575 0.3440452  0.1006814 1.4814795
## 6075ab05856f36729d2e5247 -0.358677088 0.3246317 -0.9711934 0.2677155
plot(time1, ask = FALSE)

pp_check(time1, nsamples = 150) + scale_x_log10()

time2 <- time0.with(c("scenario", "education_field"))
summary(time2)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: time ~ 1 + high_debt_version + (1 | session) + education_field + scenario 
##    Data: as.data.frame(data) (Number of observations: 44) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~session (Number of levels: 22) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.35      0.16     0.03     0.68 1.01      716      926
## 
## Population-Level Effects: 
##                                    Estimate Est.Error l-95% CI u-95% CI Rhat
## Intercept                              7.65      0.24     7.18     8.12 1.00
## high_debt_versionfalse                -0.18      0.16    -0.50     0.14 1.00
## education_fieldInteractionDesign      -0.46      0.50    -1.41     0.53 1.00
## education_fieldNone                    1.03      0.49     0.06     2.00 1.00
## education_fieldSoftwareEngineering     0.00      0.25    -0.52     0.48 1.00
## scenariotickets                       -0.33      0.16    -0.64    -0.02 1.00
##                                    Bulk_ESS Tail_ESS
## Intercept                              3071     2368
## high_debt_versionfalse                 5956     2722
## education_fieldInteractionDesign       3093     2613
## education_fieldNone                    3046     2586
## education_fieldSoftwareEngineering     2611     2244
## scenariotickets                        5745     2607
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     4.14      1.27     2.13     6.99 1.01     1164     2481
## 
## Samples were drawn using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
ranef(time2)
## $session
## , , Intercept
## 
##                             Estimate Est.Error       Q2.5     Q97.5
## 6033d69a5af2c702367b3a95 -0.08290413 0.2776848 -0.6576833 0.4794731
## 6033d90a5af2c702367b3a96  0.28846907 0.2851981 -0.1662967 0.9317805
## 6034fc165af2c702367b3a98  0.25220641 0.2666599 -0.2095262 0.8122280
## 603500725af2c702367b3a99 -0.42956787 0.3621034 -1.1625152 0.1587407
## 603f97625af2c702367b3a9d -0.12485602 0.2789997 -0.7216238 0.4199110
## 603fd5d95af2c702367b3a9e  0.18792666 0.2596450 -0.2467932 0.7802483
## 60409b7b5af2c702367b3a9f  0.36927576 0.2978414 -0.1043978 0.9972205
## 604b82b5a7718fbed181b336 -0.28050868 0.2891808 -0.8744464 0.2152571
## 6050c1bf856f36729d2e5218  0.15562791 0.2536122 -0.2993753 0.7043477
## 6050e1e7856f36729d2e5219  0.08412696 0.2548139 -0.3916256 0.6374075
## 6055fdc6856f36729d2e521b -0.04240217 0.2669613 -0.6009600 0.4917731
## 60589862856f36729d2e521f  0.02290161 0.2486071 -0.4700763 0.5317337
## 605afa3a856f36729d2e5222 -0.06100691 0.2694539 -0.6110642 0.4707718
## 605c8bc6856f36729d2e5223 -0.31263532 0.3064852 -0.9642916 0.1990548
## 605f3f2d856f36729d2e5224 -0.05251897 0.2627621 -0.5823249 0.4592440
## 605f46c3856f36729d2e5225 -0.19508397 0.2747325 -0.7662646 0.3041764
## 60605337856f36729d2e5226  0.22961749 0.2710579 -0.2181082 0.8425493
## 60609ae6856f36729d2e5228  0.30282376 0.2981207 -0.1896798 0.9568542
## 6061ce91856f36729d2e522e -0.02742233 0.2519659 -0.5454498 0.4788205
## 6061f106856f36729d2e5231 -0.28821673 0.3063634 -0.9317350 0.2390311
## 6068ea9f856f36729d2e523e  0.14713640 0.3625149 -0.5075038 0.9692353
## 6075ab05856f36729d2e5247 -0.06698761 0.3513492 -0.8305661 0.6278122
plot(time2, ask = FALSE)

pp_check(time2, nsamples = 150) + scale_x_log10()

time3 <- time0.with(c("scenario", "mo(education_level)"), edlvl_prior)
summary(time2)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: time ~ 1 + high_debt_version + (1 | session) + education_field + scenario 
##    Data: as.data.frame(data) (Number of observations: 44) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~session (Number of levels: 22) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.35      0.16     0.03     0.68 1.01      716      926
## 
## Population-Level Effects: 
##                                    Estimate Est.Error l-95% CI u-95% CI Rhat
## Intercept                              7.65      0.24     7.18     8.12 1.00
## high_debt_versionfalse                -0.18      0.16    -0.50     0.14 1.00
## education_fieldInteractionDesign      -0.46      0.50    -1.41     0.53 1.00
## education_fieldNone                    1.03      0.49     0.06     2.00 1.00
## education_fieldSoftwareEngineering     0.00      0.25    -0.52     0.48 1.00
## scenariotickets                       -0.33      0.16    -0.64    -0.02 1.00
##                                    Bulk_ESS Tail_ESS
## Intercept                              3071     2368
## high_debt_versionfalse                 5956     2722
## education_fieldInteractionDesign       3093     2613
## education_fieldNone                    3046     2586
## education_fieldSoftwareEngineering     2611     2244
## scenariotickets                        5745     2607
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     4.14      1.27     2.13     6.99 1.01     1164     2481
## 
## Samples were drawn using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
ranef(time2)
## $session
## , , Intercept
## 
##                             Estimate Est.Error       Q2.5     Q97.5
## 6033d69a5af2c702367b3a95 -0.08290413 0.2776848 -0.6576833 0.4794731
## 6033d90a5af2c702367b3a96  0.28846907 0.2851981 -0.1662967 0.9317805
## 6034fc165af2c702367b3a98  0.25220641 0.2666599 -0.2095262 0.8122280
## 603500725af2c702367b3a99 -0.42956787 0.3621034 -1.1625152 0.1587407
## 603f97625af2c702367b3a9d -0.12485602 0.2789997 -0.7216238 0.4199110
## 603fd5d95af2c702367b3a9e  0.18792666 0.2596450 -0.2467932 0.7802483
## 60409b7b5af2c702367b3a9f  0.36927576 0.2978414 -0.1043978 0.9972205
## 604b82b5a7718fbed181b336 -0.28050868 0.2891808 -0.8744464 0.2152571
## 6050c1bf856f36729d2e5218  0.15562791 0.2536122 -0.2993753 0.7043477
## 6050e1e7856f36729d2e5219  0.08412696 0.2548139 -0.3916256 0.6374075
## 6055fdc6856f36729d2e521b -0.04240217 0.2669613 -0.6009600 0.4917731
## 60589862856f36729d2e521f  0.02290161 0.2486071 -0.4700763 0.5317337
## 605afa3a856f36729d2e5222 -0.06100691 0.2694539 -0.6110642 0.4707718
## 605c8bc6856f36729d2e5223 -0.31263532 0.3064852 -0.9642916 0.1990548
## 605f3f2d856f36729d2e5224 -0.05251897 0.2627621 -0.5823249 0.4592440
## 605f46c3856f36729d2e5225 -0.19508397 0.2747325 -0.7662646 0.3041764
## 60605337856f36729d2e5226  0.22961749 0.2710579 -0.2181082 0.8425493
## 60609ae6856f36729d2e5228  0.30282376 0.2981207 -0.1896798 0.9568542
## 6061ce91856f36729d2e522e -0.02742233 0.2519659 -0.5454498 0.4788205
## 6061f106856f36729d2e5231 -0.28821673 0.3063634 -0.9317350 0.2390311
## 6068ea9f856f36729d2e523e  0.14713640 0.3625149 -0.5075038 0.9692353
## 6075ab05856f36729d2e5247 -0.06698761 0.3513492 -0.8305661 0.6278122
plot(time2, ask = FALSE)

pp_check(time2, nsamples = 150) + scale_x_log10()

Final model

All models look nice, none is significantly better than the others, we go for the simplest model. time0

Trying to add covariance

time0_with_c <- brm(
  "time ~ 1 + high_debt_version + (1 | c | session)",
  prior = c(
    prior(normal(0, 1), class = "b"),
    prior(normal(9, 1), class = "Intercept"),
    prior(exponential(1), class = "sd"),
    prior(gamma(0.01, 0.01), class = "shape")
  ),
  family = negbinomial(),
  data = as.data.frame(d.both_completed),
  control = list(adapt_delta = 0.95),
  file = "fits/time0_with_c",
  file_refit = "on_change"
)
## Start sampling
## Running MCMC with 4 parallel chains...
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## 
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## Total execution time: 1.0 seconds.
loo(
  time0,
  time0_with_c
)
## Warning: Found 4 observations with a pareto_k > 0.7 in model 'time0'. It is
## recommended to set 'moment_match = TRUE' in order to perform moment matching for
## problematic observations.
## Warning: Found 3 observations with a pareto_k > 0.7 in model 'time0_with_c'. It
## is recommended to set 'moment_match = TRUE' in order to perform moment matching
## for problematic observations.
## Output of model 'time0':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.5  7.3
## p_loo        14.5  3.0
## looic       734.9 14.6
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     28    63.6%   735       
##  (0.5, 0.7]   (ok)       12    27.3%   556       
##    (0.7, 1]   (bad)       4     9.1%   38        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Output of model 'time0_with_c':
## 
## Computed from 4000 by 44 log-likelihood matrix
## 
##          Estimate   SE
## elpd_loo   -367.4  7.0
## p_loo        14.3  2.8
## looic       734.7 14.0
## ------
## Monte Carlo SE of elpd_loo is NA.
## 
## Pareto k diagnostic values:
##                          Count Pct.    Min. n_eff
## (-Inf, 0.5]   (good)     30    68.2%   722       
##  (0.5, 0.7]   (ok)       11    25.0%   528       
##    (0.7, 1]   (bad)       3     6.8%   23        
##    (1, Inf)   (very bad)  0     0.0%   <NA>      
## See help('pareto-k-diagnostic') for details.
## 
## Model comparisons:
##              elpd_diff se_diff
## time0_with_c  0.0       0.0   
## time0        -0.1       0.4

Adding the covariance parameter did not improve the model and we will therefore continue to use the simple model time0.

Model with with incomplete data points

Some participants did only complete one scenario. Those has been excluded from the initial dataset to improve sampling of the models. We do however want to use all data we can and will therefore try to fit the model with the complete dataset.

time0.all <- brm(
  "time ~ 1 + high_debt_version + (1 | session)",
  prior = c(
    prior(normal(0, 1), class = "b"),
    prior(normal(9, 1), class = "Intercept"),
    prior(exponential(1), class = "sd"),
    prior(gamma(0.01, 0.01), class = "shape")
  ),
  family = negbinomial(),
  data = as.data.frame(d.completed),
  control = list(adapt_delta = 0.95),
  file = "fits/time0_with_all_submissions",
  file_refit = "on_change"
)
## Start sampling
## Running MCMC with 4 parallel chains...
## 
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## 
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## Mean chain execution time: 0.8 seconds.
## Total execution time: 1.3 seconds.
summary(time0.all)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: time ~ 1 + high_debt_version + (1 | session) 
##    Data: as.data.frame(d.completed) (Number of observations: 51) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Group-Level Effects: 
## ~session (Number of levels: 29) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.43      0.14     0.08     0.69 1.00      659      480
## 
## Population-Level Effects: 
##                        Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept                  7.55      0.14     7.27     7.85 1.00     3006
## high_debt_versionfalse    -0.19      0.16    -0.50     0.12 1.00     5077
##                        Tail_ESS
## Intercept                  3236
## high_debt_versionfalse     2460
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     3.73      1.07     2.01     6.16 1.00      856     1164
## 
## Samples were drawn using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
ranef(time0.all)
## $session
## , , Intercept
## 
##                             Estimate Est.Error         Q2.5     Q97.5
## 6033c6fc5af2c702367b3a93  0.10554473 0.3299606 -0.520980975 0.8257752
## 6033d69a5af2c702367b3a95 -0.12214248 0.2903288 -0.684891550 0.4616138
## 6033d90a5af2c702367b3a96  0.30962362 0.2901609 -0.212169925 0.9011978
## 6034fc165af2c702367b3a98  0.31976535 0.2937807 -0.202098775 0.9378427
## 603500725af2c702367b3a99 -0.51727261 0.3706944 -1.240419250 0.1545609
## 603f84f15af2c702367b3a9b -0.37633878 0.4171061 -1.260012750 0.3646146
## 603f97625af2c702367b3a9d -0.18443082 0.3072241 -0.788130275 0.4328387
## 603fd5d95af2c702367b3a9e  0.23937677 0.2865087 -0.283012025 0.8468931
## 60409b7b5af2c702367b3a9f  0.37604443 0.2966935 -0.148027175 1.0035638
## 604b82b5a7718fbed181b336 -0.34036066 0.3223859 -0.988829625 0.2824985
## 604f1239a7718fbed181b33f -0.03331665 0.3327295 -0.693844375 0.6188374
## 6050c1bf856f36729d2e5218  0.11978848 0.2873252 -0.415460925 0.7291963
## 6050e1e7856f36729d2e5219  0.07766598 0.2839507 -0.455587650 0.6588954
## 6055fdc6856f36729d2e521b -0.07870878 0.2844104 -0.633388100 0.5047004
## 60579f2a856f36729d2e521e -0.05188545 0.3382762 -0.721149050 0.6182157
## 60589862856f36729d2e521f  0.04593352 0.2744286 -0.464283825 0.6122181
## 605a30a7856f36729d2e5221 -0.20035656 0.3578350 -0.929110475 0.4930111
## 605afa3a856f36729d2e5222 -0.08047121 0.2897261 -0.652585400 0.5129167
## 605c8bc6856f36729d2e5223 -0.36982264 0.3290852 -1.033163250 0.2255860
## 605f3f2d856f36729d2e5224 -0.12067704 0.2869245 -0.681740025 0.4532654
## 605f46c3856f36729d2e5225 -0.25740083 0.3133558 -0.886684375 0.3404050
## 60605337856f36729d2e5226  0.23478995 0.2876444 -0.279596500 0.8412008
## 60609ae6856f36729d2e5228  0.32207713 0.2830570 -0.182777825 0.9095830
## 6061ce91856f36729d2e522e -0.03948155 0.2835850 -0.600204050 0.5271590
## 6061f106856f36729d2e5231 -0.38125561 0.3349124 -1.032329750 0.2360673
## 60672faa856f36729d2e523c -0.13173230 0.3378698 -0.814172800 0.5648111
## 6068ea9f856f36729d2e523e  0.67996973 0.3365452  0.004518002 1.3390048
## 606db69d856f36729d2e5243  0.55522922 0.3677054 -0.050337110 1.3325688
## 6075ab05856f36729d2e5247 -0.30790146 0.3239894 -0.950151450 0.3000961
plot(time0.all, ask = FALSE)

pp_check(time0.all, nsamples = 150) + scale_x_log10()

Training the model on all data points reduces the uncertainty and did not result in any sampling problems. We will proceed with the model fitted to all the data.

Interpreting the model

Extract posterior samples:

post <- posterior_predict(time0.all, newdata = data.frame(high_debt_version = c("false", "true"), session = NA))
post.low <-  post[,1]
post.high <- post[,2]
post.diff <- post.high - post.low
post.diff.scaled <- post.diff / mean(post)

When we look at the distribution of the data between the high and low debt version we can see a difference, they are however quite similar.

summary(data.frame("Low Debt"=post.low, "High Debt"=post.high))
##     Low.Debt       High.Debt    
##  Min.   :   11   Min.   :    6  
##  1st Qu.:  830   1st Qu.: 1024  
##  Median : 1376   Median : 1646  
##  Mean   : 1710   Mean   : 2066  
##  3rd Qu.: 2213   3rd Qu.: 2664  
##  Max.   :13297   Max.   :18839
ggplot(data.frame(time = post.high)) +
  geom_density(data = data.frame(time = post.low), aes(x=time/60, colour = "blue")) +
  geom_density(data = data.frame(time = post.high),  aes(x=time/60, colour = "red" )) +
  scale_x_log10() +
  labs(
    title = "Posterior density of time for high and low debt versions",
    subtitle = "Notice! x-axis is log10 scaled.",
    x ="Time (min)",
    y = "Density"
  ) +
  theme_minimal() +
  scale_y_continuous(breaks = NULL) +
  scale_colour_manual(name="Version",labels=c("Low Debt", "High Debt"), values=c("blue", "red"))

We can also plot the difference between time for the high debt version and the low debt version. In those graphs we see that the difference is centered around zero with slightly more weight on the positive (indicating slightly longer times for the high debt version)

ggplot(data.frame(x = post.diff/60)) + geom_density(aes(x=x)) +
  labs(
    title = "Posterior density of time difference low and high debt versions ",
    subtitle = "Difference = High Debt time - Low Debt time",
    x ="Time Difference (min)",
    y = "Density"
  ) +
  theme_minimal() +
  scale_y_continuous(breaks = NULL)

data.frame(x = post.diff.scaled) %>%
  ggplot() +
  geom_density(aes(x=x)) +
  labs(
    title = "Posterior density of time difference low and high debt versions ",
    subtitle = "Difference = High Debt time - Low Debt time",
    x ="Time Difference (prortion of mean time)",
    y = "Density"
  ) +
  theme_minimal() +
  scale_y_continuous(breaks = NULL)

We can also check the probability of the high debt effect being less than as well as grather then zero.

sprintf("high debt effect < 0: %.2f%%", sum(sign(post.diff) == -1)/length(post.diff) * 100)
## [1] "high debt effect < 0: 39.83%"
sprintf("high debt effect > 0: %.2f%%", sum(sign(post.diff) == 1)/length(post.diff) * 100)
## [1] "high debt effect > 0: 60.15%"

We can see no “significant” relation between the amount of technical debt in the given codebase and time spent on the task.

---
title: "Time Outcome"
author: Hampus Broman & William Levén
date: 2021-04
output: 
  html_document: 
    pandoc_args: [ "-o", "html/time.html" ]
---

```{r include-setup, include=FALSE}
source(knitr::purl('setup.Rmd', output = tempfile()))
```


## Looking at the data
We plot the data and can see that there is no obvious large difference between the debt versions
```{r}
d.both_completed %>%
  ggplot(aes(x=time/60, fill=high_debt_version)) + 
  geom_boxplot() +
  labs(
    title = "Distribution of time measurements for the different debt levels",
    subtitle = "Notice! Log10 x-scale",
    x ="Time (min)"
  ) +
  scale_y_continuous(breaks = NULL) +
  scale_x_log10() +
  scale_fill_discrete(name = "Debt Level", labels = c("High Debt", "Low Debt"), guide = guide_legend(reverse = TRUE)) +
  theme_minimal()
```

## Descriptive Statistics
For the whole dataset:
```{r}
d.both_completed %>%
  pull(time) %>% 
  summary()

variance <- d.both_completed %>%
  pull(time) %>%
  var()
sprintf("Variance: %.0f", variance)
```


## Initial model
As the variance is much greater than the mean we will use a negative binomial family that allows us to model the variance separately.

We include `high_debt_verison` as well as a varying intercept for each individual in our initial model.

### Selecting Priors
We iterate over the model until we have sane priors
```{r}

time0.with <- extendable_model(
  base_name = "time0",
  base_formula = "time ~ 1 + high_debt_version + (1 | session)",
  base_priors = c(
    prior(normal(0, 1), class = "b"),
    prior(normal(7.5, 1), class = "Intercept"),
    prior(exponential(1), class = "sd"),
    prior(gamma(0.01, 0.01), class = "shape")
  ),
  family = negbinomial(),
  data = d.both_completed,
  base_control = list(adapt_delta = 0.95)
)

# Default priors:
prior_summary(time0.with(only_priors= TRUE))

# Our priors:
prior_summary(time0.with(sample_prior = "only"))

# Prior predictive check
pp_check(time0.with(sample_prior = "only"), nsamples = 200) + 
  scale_x_log10()
```
### Model fit
We check the posterior distribution and can see that the model seems to have been able to fit the data well
```{r}
# Posterior predictive check
pp_check(time0.with(), nsamples = 100) +
  scale_x_log10()

summary(time0.with())
```

## Model extenstions

We use loo to check some possible extensions on the model.

```{r}

edlvl_prior <- prior(dirichlet(2), class = "simo", coef = "moeducation_level1")

loo(
  time0.with(),
  time0.with("work_domain"),
  time0.with("work_experience_programming.s"),
  time0.with("work_experience_java.s"),
  time0.with("education_field"),
  time0.with("mo(education_level)", edlvl_prior),
  time0.with("workplace_peer_review"),
  time0.with("workplace_td_tracking"),
  time0.with("workplace_pair_programming"),
  time0.with("workplace_coding_standards"),
  time0.with("scenario"),
  time0.with("group")
)
```


Some variables seem to stand out as better than other. We try combining those to see if we can further improve our model.

```{r}
loo(
  time0.with(),
  time0.with("scenario"),
  time0.with("education_field"),
  time0.with("workplace_peer_review"),
  time0.with("mo(education_level)", edlvl_prior),
  time0.with(c("scenario", "education_field")),
  time0.with(c("scenario", "mo(education_level)"), edlvl_prior),
  time0.with(c("scenario", "workplace_peer_review")),
  time0.with(c("education_field", "mo(education_level)"), edlvl_prior),
  time0.with(c("education_field", "workplace_peer_review")),
  time0.with(c("mo(education_level)", "workplace_peer_review"), edlvl_prior)
)
```
None one of the new models stand out as particularly better than the old models, We try combining the so far most prominent features in one model.

```{r}
loo(
  time0.with(),
  time0.with("scenario"),
  time0.with(c("scenario", "education_field")),
  time0.with(c("scenario", "mo(education_level)"), edlvl_prior),
  time0.with(c("scenario", "mo(education_level)", "education_field"), edlvl_prior)
)
```

## Candidate models
We inspect some of our top performing models. 

All models seems to have sampled nicely (rhat = 1 and fluffy plots) they also have about the same fit to the data end similar estimated for the high_debt_version beta parameter


```{r}
time0 <- time0.with()
summary(time0)
ranef(time0)
plot(time0, ask = FALSE)

pp_check(time0, nsamples = 150) + scale_x_log10()

```

```{r}
time1 <- time0.with("scenario")
summary(time1)
ranef(time1)
plot(time1, ask = FALSE)

pp_check(time1, nsamples = 150) + scale_x_log10()
```

```{r}
time2 <- time0.with(c("scenario", "education_field"))
summary(time2)
ranef(time2)
plot(time2, ask = FALSE)

pp_check(time2, nsamples = 150) + scale_x_log10()
```

```{r}
time3 <- time0.with(c("scenario", "mo(education_level)"), edlvl_prior)
summary(time2)
ranef(time2)
plot(time2, ask = FALSE)

pp_check(time2, nsamples = 150) + scale_x_log10()
```

### Final model
All models look nice, none is significantly better than the others, we go for the simplest model. time0

### Trying to add covariance

```{r}
time0_with_c <- brm(
  "time ~ 1 + high_debt_version + (1 | c | session)",
  prior = c(
    prior(normal(0, 1), class = "b"),
    prior(normal(9, 1), class = "Intercept"),
    prior(exponential(1), class = "sd"),
    prior(gamma(0.01, 0.01), class = "shape")
  ),
  family = negbinomial(),
  data = as.data.frame(d.both_completed),
  control = list(adapt_delta = 0.95),
  file = "fits/time0_with_c",
  file_refit = "on_change"
)
```

```{r}
loo(
  time0,
  time0_with_c
)
```
Adding the covariance parameter did not improve the model and we will therefore continue to use the simple model time0.

### Model with with incomplete data points

Some participants did only complete one scenario. Those has been excluded from the initial dataset to improve sampling of the models. We do however want to use all data we can and will therefore try to fit the model with the complete dataset.

```{r}
time0.all <- brm(
  "time ~ 1 + high_debt_version + (1 | session)",
  prior = c(
    prior(normal(0, 1), class = "b"),
    prior(normal(9, 1), class = "Intercept"),
    prior(exponential(1), class = "sd"),
    prior(gamma(0.01, 0.01), class = "shape")
  ),
  family = negbinomial(),
  data = as.data.frame(d.completed),
  control = list(adapt_delta = 0.95),
  file = "fits/time0_with_all_submissions",
  file_refit = "on_change"
)
summary(time0.all)
ranef(time0.all)
plot(time0.all, ask = FALSE)

pp_check(time0.all, nsamples = 150) + scale_x_log10()
```

Training the model on all data points reduces the uncertainty and did not result in any sampling problems. We will proceed with the model fitted to all the data.

## Interpreting the model

Extract posterior samples:
```{r}

post <- posterior_predict(time0.all, newdata = data.frame(high_debt_version = c("false", "true"), session = NA))
post.low <-  post[,1]
post.high <- post[,2]
post.diff <- post.high - post.low
post.diff.scaled <- post.diff / mean(post)

```


When we look at the distribution of the data between the high and low debt version we can see a difference, they are however quite similar.
```{r}
summary(data.frame("Low Debt"=post.low, "High Debt"=post.high))

ggplot(data.frame(time = post.high)) +
  geom_density(data = data.frame(time = post.low), aes(x=time/60, colour = "blue")) +
  geom_density(data = data.frame(time = post.high),  aes(x=time/60, colour = "red" )) +
  scale_x_log10() +
  labs(
    title = "Posterior density of time for high and low debt versions",
    subtitle = "Notice! x-axis is log10 scaled.",
    x ="Time (min)",
    y = "Density"
  ) +
  theme_minimal() +
  scale_y_continuous(breaks = NULL) +
  scale_colour_manual(name="Version",labels=c("Low Debt", "High Debt"), values=c("blue", "red"))

```

We can also plot the difference between time for the high debt version and the low debt version. In those graphs we see that the difference is centered around zero with slightly more weight on the positive (indicating slightly longer times for the high debt version)
```{r}

ggplot(data.frame(x = post.diff/60)) + geom_density(aes(x=x)) +
  labs(
    title = "Posterior density of time difference low and high debt versions ",
    subtitle = "Difference = High Debt time - Low Debt time",
    x ="Time Difference (min)",
    y = "Density"
  ) +
  theme_minimal() +
  scale_y_continuous(breaks = NULL)


data.frame(x = post.diff.scaled) %>%
  ggplot() +
  geom_density(aes(x=x)) +
  labs(
    title = "Posterior density of time difference low and high debt versions ",
    subtitle = "Difference = High Debt time - Low Debt time",
    x ="Time Difference (prortion of mean time)",
    y = "Density"
  ) +
  theme_minimal() +
  scale_y_continuous(breaks = NULL)

```


We can also check the probability of the high debt effect being less than as well as grather then zero.
```{r}
sprintf("high debt effect < 0: %.2f%%", sum(sign(post.diff) == -1)/length(post.diff) * 100)
sprintf("high debt effect > 0: %.2f%%", sum(sign(post.diff) == 1)/length(post.diff) * 100)
```

We can see no "significant" relation between the amount of technical debt in the given codebase and time spent on the task.

